1 2
Sharad Chandra Shukla , * Agraj Tripathi
1 2
Research Scholar,Sai Nath University, Ranchi, Jharkhand,India, Bhabha Institute of Technology, Kanpur, U. P. India
Shape design and representation of complex objects is a hard key problem in the industrial core domain of computational mathematics applications. A work piece must be represented in some standard complex object description format such that its representation can be efficiently used in a complex process like redesign. To that end, a digitizing process represents the object surface as a weakly structured discrete and digitized set of 3D points. Surface design attempts to transform this representation into an efficient mathematics representation. The paper presents functions of complex topology and complex surface geometry redesign of complex surface using complex mathematics.
Keywords: Complex design; Complex topology; Complex surface; Complex geometry
ABSTRACT
INTRODUCTION
Modeling as the name sometimes synonymous with geometrical modeling is rapidly emerging as a central area of research and development in many mathematical applications. All these applications require representing the shapes of Solid Complex physical objects and such representations and basic operation on them can be provided by Solid Complex modeling. That can model certain classes of piecewise parametric surfaces. Geometric or surface modeling traditionally identifies a body of techniques that can model certain classes of piecewise parametric surfaces, subject to particular conditions of shape and smoothness. It developed as a separate field in several industries, including automobile, aerospace, and shipbuilding, it has some of its intellectual roots in approximation theory. It is our view that the streams of geometric and Solid Complex modeling are converging. As Solid Complex modeling strives to extend the geometric coverage, there is an emerging need to research the use of surface forms and the techniques to interrogate them. Similarly, as geometric modeling contemplates building complete Solid Complex representations from surface patches, the usefulness of traditional Solid Complex-modeling techniques is more widely recognized.
Surface design that is the automatic construction of a mathematics object from data is a hard and industrially relevant problem. The task being considered in this paper is the design from a given 3D point data set. The
problem core is that, any given set represents infinitely many different geometrical surfaces, that is those and only those surfaces that have the set in common. However, the data set represents only one physical surface, which is the surface of that physical object from which a digitizing process generated the data set. Thus, a surface-design system must reconstruct a mathematics object that approximates the physical object. This corresponds to the task of recognizing a physical object in a 3D point set, which is a special case of pattern recognition. The system must perform this task such that a construction engineer can start working with the resulting mathematics object without being forced to introduce an expensive manual modification to the representation.
LITERATURE REVIEW
Various efforts have been made to Shape design a three-dimensional object's structure (geometrical, topological and unstructured). Till date many function/formulas have been proposed for the unstructured surface Shape design. An intense discussion on categorization of irregular surfaces is going on in academics and R&D community. Although it is yet to develop best fit function/formula for the Shape design of unstructured three dimensional surfaces. This paper presents a survey conducted on the valuable efforts made by the researchers in the field of Shape design of surfaces.
*Address for correspondence: Dr. Agraj Tripathi, Associate Professor, Bhabha Institute of Technology, Kanpur, U. P. India; Email ID: [email protected]
Partition of unity (POU) method is to divide the global [3]
domain into a sub domain where the problem can be solved locally. More formally, the global difficult problem p is decomposed into several smaller local problem p and their local solutions S are combined i i together using the weighting coefficients w (p) of S that i i act as gluing functions to obtain a global solution S. Consider a global domain and divide it into M “slightly” overlapping sub domains
Consider a global domain and divide it into M “slightly” overlapping sub domains
Where i
By this local Shape design functions f can be computed i Here two families of the functions have to be built
(i) The weighting functions W andi (ii) The local Shape design function f i
The above both Shape design functions are using only for local domain. This local domain function is capable to capturing the local shape of surface.
[4]
The exact geodesic function/formula has been used by a simple parameterization of the distance function over the edges; the implementation is actually practical even though, the best of knowledge, it is never been
[ 5 ]
d o n e p r e v i o u s l y . We c a n s e e t h a t t h e 2 function/formula's worst case running time of O (n log n ) i s p e s s i m i s t i c , a n d t h a t i n p r a c t i c e t h e function/formula runs in sub-quadratic time. For instance, we can compute the exact geodesic distance from a point to all the vertices of a 400K-triangle mesh in about one minute. The basic idea of the MMP (Mitchell Mount and Papadimitriou) function/formula is to track together groups of shortest paths that can be parameterized automatically. This is achieved by partitioning each mesh edge into a set of intervals that we call surface's and show that all the shortest paths with in a surface can be encoded locally using a 6 tuple (b ,b ,d ,d ,σ ,τ) .The surfaces are then propagated 0 1 0 1 across faces of the mesh in Dijkstra-like sweep. The distance eld D (p) over the surface is expressed as a tuple (b , b , d d , σ, τ) where b ,b define the endpoints 0 1 0, 1 0 1 of w, d ,d are the corresponding distances to the 0 1 pseudo source, σ is the geodesic distance from s to the
source v , and τ encodes the direction of s from the s directed edge e.
TOPOLOGICAL SURFACE
A classic surface-design method uses the idea of constructing an approximating surface with a usually very large number of plane pieces. Triangulation, for instance, yields an approximating surface consisting of triangles as plane pieces. In this context, an intuitive idea of smoothness of the approximating surface is used: the surface is considered “smooth” in a certain area if the angles between the area's plane pieces are not “too wide”. The formalized search for a smooth triangulation of a point coordinate is hard. Different [8, methods yielding smooth triangulations can be found 9, 10, 11]
.
There are several topological properties of a surface that may be used by an advanced classic surface-design method. In order to determine the peculiar properties of a given surface, a gridded triangulation may be calculated from a smooth triangulation. A triangulation is called “gridded” if and only the triangle corners are in the normal vectors of the points of a uniform 2D grid. The grid represents a physical plane area on which the physical object rests during the digitizing process. A grid point represents a point that is aimed at by a sensor, like a pin of a tactile digitizing hardware. The sensor aims at this grid point along a vector that is orthogonal to the plane area. For a gridded triangulation, the indicated topological properties can be easier determined than for a non-gridded triangulation. Two examples of such properties are a “normal vector” and
[12] “Gaussian curvature” .
POLYNOMIAL LAPLACIAN
A two-dimensional, three-dimensional, polynomial curve with Laplace parameterized curve
2 3
Where a0, ,……a1 an1, R , and R an
Missing surface may be regular and irregular both types, it is easy to say that the surface may be repaired either to be surface patch or to be extended boundary curves according to grid projection. Parametric curves can be extended
[3] with the help of chain rule (CR) .
(3)
If f(x , x , x ) is a parametric function defined as f (x , x , x ) = {f (x , x , x ), f (x , x , x ), f (x , x , x )} R . If x1 2 3 1 2 3 1 1 2 3 2 1 2 3 3 1 2 3 1
x (t) and x2 x (t) and x x (t).
= 1 = 2 3 = 3
(3)
Where x (t), x (t), x (t) R then 1 2 3
Further if x = x (u , u , u ) and x = (u , u , u ) and x = (u , u , u ) 1 1 2 3 2 1 2 3 3 1 2 3 Where x (u , u , u ), x (u , u , u ), x (u , u , 1 1 2 3 2 1 2 3 3 1 2
(3).
R u ) are the control points of curve.3
s called Jacobian matrix of reparameterizations. Its determined is called the Jacobian, of the transformation.
Laplace Function Tolerance of model Purification of Node
L( ) 7.5 .02
x1= x (u1, u2, u3) 12.5 .01
x2 = (u1, u2, u3) 13 .01
x3 = (u1, u2, u3) 14 .05
Table -1 Tolerance of mathematical model
To use of chain rule, jacobian surface patch (JSP) can be developed through a given parameters
CONCLUSION
The present results show the correct and consistent representations of three-dimensional objects are required by applications as varied as modeling, simulation, visualization, mathematics and finite element analysis. However, most acquired 3D models, whether created by hand or by using automatic tools, contain errors and inconsistencies. The result represents the tolerance of Laplace function use by the given parameter with specific mathematic formula. REFERENCES
1. Armin Iske, Ewald Quak, Michele floater. Scatter data modeling using radial basis function
2. Banzhaf, W. (1997), “Interactive evolution,” in Handbook of Evolutionary Computation, Z. M. T. Back, D. Fogel (Ed.), pp C2.9,1–7, Oxford University Press.
3. Banzhaf, W., Nordin, P., Keller, R., and Francone, F. (1998), Genetic Programming - an Introduction; on the Automatic Evolutionof Computer Programs and Its Application, dpunkt-Verlag, Heidelberg. Morgan Kaufmann, San Francisco.
4. Barequet G and Mocha Shair. Filling Gaps in the boundary of a polyhedron. CAGD March 1995 5. Piegl, L. and Tiller, W. (1997), the NURBS Book,
Springer. Project team SURREAL (1998),
“Fl¨achenrekonstruktion mit Genetischem Programmieren,” Technical report, University of Dortmund, Computer Science Department, Dortmund, Germany.
6. L. Piegl, "On NURBS: A Survey, "IEEE CG&A, Vol. 11, No. 1, Jan. 1991, pp. 55-71. Citation: Gerald Farin
7. Piegl L., Tiller W., Curve and surface constructions using rational B-splines, Computer Aided Design, Volume19, Number-9, 1987,pp.485-498.
8. Prabir Bhattacharya, Azriel Rosenfield “Convexity Property of space curves” ScienceDirect, Dec- 2002
9. Praun E., McLennan T.J. Smooth surface reconstruction from noisy range data. In Proceedings of Graphite 2003, pages 119–126, 10. Qinmin D.X., Tian, Z.X., Zhang, Y.X., Feng J.B.,
The method of numeric polish in curve fitting, ACTA MATHEMATICA SINICA, 1975, 18(3): 173-184. (in Chinese)
11. Rechenberg I. (Ed.) (1994), Evolutionsstrategie ' 9 4 , v o l u m e 1 o f We r k s t a t t B i o n i k u n d Evolutionstechnik, Stuttgart, Frommann- Holzboog.
12. Reed, K., Kelly, J. C., Harrod, D., and Conroy, W. ( 1 9 9 1 ) , t h e I n i t i a l G r a p h i c s E x c h a n g e Specification (IGES) Version 5.1., Fairfax, VA: N a t i o n a l C o m p u t e r G r a p h i c s A s s o c . , Administrator-IGES/PDES Organization